Abstract

Actuarial reserving deals with the problem of predicting outstanding claims payments on policies issued up to today to find an appropriate amount of capital, the claims reserve or technical provisions, to set aside in order to be able to meet obligations to policyholders. Historically, and commonly still today, this has been approached using purely algorithmic and deterministic methods, not based in any statistical models. This thesis contains five individual papers, mainly concerned with statistical models for use in the area of reserving in non-life insurance.

Paper I sets out all the components needed for the valuation of aggregate non-life insurance liability cash flows based on data in the form of claims triangles. The paper contains all necessary ingredients for use in practice, including the estimation of model parameters and a bias correction of the plug-in estimator of the valuation formula. The valuation framework that the paper takes as its starting point is compatible with the view of the Solvency IIdirective on how to compute the value of the technical provisions, i.e. that the value should equal the amount which a so-called reference undertaking would demand in order to take over and handle the run-off of the liability cash flow.

Paper II deals with the problem of estimating the conditional mean squared error of prediction(MSEP), conditional on the observed data. The paper presents an approach that yields analytically computable estimators for a wide range of different models — otherwise readily computable using simple numerical methods — and, moreover, it shows that the approach reproduces the famous MSEP formula for the distribution-free chain ladder model given by Mack in 1993. The approach is particularly useful when considering run-off triangles since itis then not feasible to perform a prediction assessment based on out-of-sample performance.

Paper III is concerned with properties of the variance of the variance parameter estimator ina general linear model, mainly in the form of finite sample size bounds that are independent of the covariates and that are such that, asymptotically, the lower and upper bounds are the same. As opposed to the other papers of this thesis, this paper is purely theoretical without an immediate insurance context — except for a small example.

Paper IV introduces a discrete-time micro-model called the collective reserving model (CRM). The model is highly accessible since, even though it is a micro-model, it is modelled on the aggregate level using two triangles, one for the number of reported claims and one for the claims payments. The paper shows, among other things, how the model gives predictors of outstanding claims payments separately for incurred but not reported and reported but not settled claims, and, interestingly, shows that the chain ladder technique is a large exposure (e.g. the number of contracts) approximation of the CRM.

Paper V is chiefly concerned with deriving closed-form expressions for moments in a class of continuous-time micro-models. It is the first paper to accomplish this task, hopefully making continuous-time micro-models accessible to a broader audience.